Footing 350x350, 8-16 mm bars, reversible moment

Data and assumptions

• Column 350 mm × 350 mm (long side $a=350$ mm along the footing length); footing concrete M20, steel Fe500; column concrete M30; column bars 8-16 mm.
• Given loads are factored: $P_u = 1100$ kN, $M_u = 60$ kN·m. Service loads = factored / 1.5: $P = 733.3$ kN, $M = 40.0$ kN·m.
• Allowable (safe) bearing capacity of soil = 150 kN/m² at 1.5 m depth.
• The moment is reversible: same steel on both sides.
• Clear cover 50 mm for the footing (IS 456 cl. 26.4.2.2); effective depth $d = D - 60$ mm (cover + half bar). Load factor 1.5.

Size of footing

Weight of footing and backfill is taken as 10% of the column load.

$$A_{req} = \frac{1.1\,P}{q_a} = \frac{1.1 \times 733.3}{150} = 5.38\ \text{m}^2$$

Eccentricity $e = M/P = 40.0/733.3 = 0.055$ m, which is less than $L/6$, so there is no tension under the base. The size is chosen so that

$$q_{max} = 1.1\frac{P}{BL}\left(1 + \frac{6e}{L}\right) \le q_a$$

Try $L \times B = 3.3 \times 1.8$ m ($A = 5.94$ m²): $q_{max} = 149.3$ kN/m² $\le 150.0$ and $q_{min} = 122.3$ kN/m² $> 0$. Safe.

Factored soil pressure

$P_u = 1.5 \times 733.3 = 1100.0$ kN, $M_u = 60.0$ kN·m. $p_u = \dfrac{P_u}{BL}\left(1 \pm \dfrac{6e}{L}\right)$: $p_{u,max} = 203.6$ kN/m², $p_{u,min} = 166.8$ kN/m² (average 185.2 kN/m²). Self-weight does not cause bending in the footing, so it is excluded.

Bending moments and reinforcement

Cantilever projections from the column face: along $L$: $(3300 - 350)/2 = 1475$ mm; along $B$: $(1800 - 350)/2 = 725$ mm. Along $L$ (critical section at the column face on the side of maximum pressure, linear pressure): $M_{u,L} = B\left[p_f\dfrac{x^2}{2} + (p_{max}-p_f)\dfrac{x^2}{3}\right]$, with $p_f = 187.1$ kN/m² at the face and $x = 1.475$ m: $M_{u,L} = 387.8$ kN·m. Along $B$ (average pressure): $M_{u,B} = 160.6$ kN·m. Required depth for flexure: $M_{u,lim} = 0.133 f_{ck} b d^2$ is far larger than the moment for $d = 640$ mm, so the depth is governed by shear (one-way or punching). Trial $D = 700$ mm, $d = 640$ mm, found by the shear checks below.

$A_{st} = \dfrac{0.5 f_{ck}}{f_y}\left[1 - \sqrt{1 - \dfrac{4.6 M_u}{f_{ck} b d^2}}\right] b d$; minimum steel = 0.12% of $bD$.

For the rectangular footing, the short-direction steel is placed in a central band of width $B = 1800$ mm in the proportion $\dfrac{2}{\beta+1} = \dfrac{2}{1.83+1} = 0.706$ (IS 456 cl. 34.3.1), where $\beta = L/B$.

• Central band: $0.706 \times 2772 = 1957$ mm² → 12 mm @ 100 mm c/c (18 bars).
• Each outer strip (750 mm wide): remaining steel 408 mm² or minimum 0.12% = 630 mm², whichever is more → 12 mm @ 120 mm c/c (6 bars).

Check for one-way shear (at distance $d$ from the column face)

• Along $L$ side: $V_u = 299.0$ kN, $\tau_v = \dfrac{V_u}{B d} = 0.260$ MPa; $p_t = 0.136\%$, $\tau_c = 0.288$ MPa (IS 456 Table 19, SP 16 formula). $\tau_v < \tau_c$, safe.
• Along $B$ side: $V_u = 51.9$ kN, $\tau_v = 0.025$ MPa; $p_t = 0.136\%$, $\tau_c = 0.288$ MPa. Safe.

Check for two-way (punching) shear (at $d/2$ from the column face)

• Perimeter $b_o = 2[(a + d) + (c + d)] = 2[(350+640) + (350+640)] = 3960$ mm
• $V_u = P_u - p_u (a+d)(c+d) = 1100.0 - 185.2 \times 0.990 \times 0.990 = 918.5$ kN (average pressure; the effect of moment transfer is neglected).
• $\tau_v = \dfrac{V_u}{b_o d} = 0.362$ MPa; $k_s = 0.5 + \beta_c = 1.00$, $\tau_c' = k_s \times 0.25\sqrt{f_{ck}} = 1.118$ MPa. $\tau_v < \tau_c'$, safe.

Development length

$L_d = \dfrac{0.87 f_y \phi}{4\tau_{bd}}$, $\tau_{bd} = 1.92$ MPa (M20, deformed bars): 12 mm bars (both directions) → $L_d = 680$ mm. Available length from column face (less 50 mm end cover): 1425 mm along $L$ (adequate), 675 mm along $B$. Along $B$ the available 675 mm is slightly less than $L_d = 680$ mm, so the bars are given 90° bends at the ends (each bend adds an anchorage value of $8\phi = 96$ mm), which is adequate.

Bearing stress and transfer of load from column

Column area $A_2 = 350\times350 = 0.122$ m². Bearing stress $= P_u/A_2 = 8.98$ MPa. Permissible bearing stress: column concrete $0.45 f_{ck} = 13.50$ MPa; footing concrete $0.45 f_{ck}\sqrt{A_1/A_2}$ with $\sqrt{A_1/A_2} \le 2$ gives up to 18.00 MPa. The lesser, 13.50 MPa, governs. Capacity of bearing = 1654 kN $\ge$ $P_u = 1100$ kN, so the whole load can be transferred by bearing. Provide minimum dowels of 0.5% of column area = 612 mm²: use 8 bars of 16 mm (1608 mm²) as dowels, same as column bars. Dowel length in compression $L_{dc} = \dfrac{0.87 f_y\phi}{4 \times 1.25\,\tau_{bd}} = 725$ mm (and the same lap with the column bars). Available depth in footing = 700 - 50 - 24 = 626 mm $< L_{dc}$; dowels are bent at the bottom with 90° bends.

Detailing

PLAN                      ELEVATION
+------------------+      col
| #  #  #  #  #  # |      | |
| #  +------+  #  # |     _|_|__
| #  | col  |  #  # |    |_____|  <- D = 700 mm
| #  +------+  #  # |    bars both ways
| #  #  #  #  #  # |
+------------------+
 L = 3.3 m, B = 1.8 m

• Bottom mesh: 12 mm @ 130 c/c along $L$ (lower layer); along $B$ (upper layer), 12 mm @ 100 c/c in the central 1.8 m band and 12 mm @ 120 c/c in each outer strip, 50 mm clear cover; dowels 8-16 mm.

Answer: Footing 3.3 m × 1.8 m × 700 mm thick; 12 mm @ 130 c/c along the length; along the width, 12 mm @ 100 c/c in the central 1.8 m band and 12 mm @ 120 c/c in the outer strips.